Math 321  Real Variables II  Spring 2015
Instructor: Malabika Pramanik
Office: 214 Mathematics Building
Email: malabika at math dot ubc dot ca
Lectures: Mon,Wed,Fri 9:00 AM to 10:00 AM in Room 104 Mathematics Building.
Office hours: Mon 1011, Wed 1112 or by appointment.
Course information
Piazza links
Homework
Practice problems
Here is a list of exercises from the textbook based on the weekly lecture material. They are not meant to be turned in for grading, but it is recommended that you work through them to get a better understanding of the topics covered. Some midterm and final exam problems may be modelled on these exercises. Unlike homework problems, detailed solutions of these problems will not be posted, but I am happy to include hints upon request.
 Sequences and series of functions
 Chapter 7: Exercises 126.For problem 22, assume that f is Riemann integrable instead of RiemannStieltjes integrable for now.
 More practice problems on Chapter 7.
 RiemannStieltjes integration
 Fourier series
 Chapter 8, Exercises 1219.
 More exercises in the practice problem sheet for the final. See at the bottom of this page.
Real analysis lecture notes on the web
Here is a list of online lecture notes of similar courses offered at various institutions.
MIT open courseware .
John Lindsay Orr's Analysis Webnotes, University of Nebraska, Lincoln.
Eric Sawyer's lecture notes ,McMaster University.
Vern Paulsen's lecture notes, University of Houston.
Lee Larson's lecture notes, University of Louiville.
An introduction to real analysis by William Trench.
Weekbyweek course outline
Here is a rough guideline of the course structure, arranged by week. The textbook pages are mentioned as a reference and as a reading guide. The treatment of these topics in lecture may vary somewhat from that of the text.
 Week 1 (pages 143154 of the textbook):
 Pointwise and uniform convergence
 Definitions and examples
 Interchanging limits.
 Week 2 (pages 143154 of the textbook):
 Applications of pointwise and uniform convergence:
 Weierstrass Mtest
 A continuous but nowhere differentiable function
 A spacefilling curve
 Separability of the space of continuous functions on [0,1]
 Week 3 (pages 143154 of the textbook):
 Density of polygonal functions in the space C[0,1]
 A quick review of uniform continuity
 Bernstein's proof of the Weierstrass approximation theorem
 Week 4 (pages 159161 of the textbook):
 Approximation of continuous periodic functions by trigonometric polynomials
 Algebras and lattices
 The StoneWeierstrass theorem  real version
 Week 5 (pages 154165 of the textbook):
 StoneWeierstrass theorem  complex version (a reading exercise)
 Review of compactness in general metric spaces
 Equicontinuity
 Statement of the ArzelaAscoli theorem
 Week 6 (pages 120130 of the textbook) :
 Proof of the ArzelaAscoli theorem
 Towards RiemannStieltjes integral
 The RiemannStieltjes integral
 Riemann's condition for RiemannStieltjes integrability
 The space of RiemannStieltjes integrable functions
 Week 7 :
 Week 8 :
 RiemannStieltjes integrability and uniform convergence
 Midterm review.
 Week 9 (pages 120133 of the textbook):
 Functions of bounded variation
 Jordan's theorem
 General integrators
 Integration by parts
 Integrators of bounded variation
 Linear functionals on normed linear spaces
 Week 10 (pages 185192 of the textbook) :
 Riesz representation theorem for continuous linear functionals on C[a,b]
 Fourier series
 Partial Fourier sums as L2 projections
 Week 11 (pages 185192 of the textbook) :
 Bessel's inequality
 Plancherel's theorem
 L2 convergence of Fourier series
 L2 approximation by continuous periodic functions
 Dirichlet's formula
 Week 12 (pages 185192 of the textbook):
 Properties of the Dirichlet kernel
 Uniform convergence, or lack thereof, of partial Fourier sums
 Fejer kernel and Cesaro summability of Fourier series
 A brief introduction to approximations to the identity
 Week 13 :
 Shortcomings of Riemann integration  some workarounds
 Sets of Lebesgue measure zero
 Lebesgue's condition on Riemann integrability
Worksheets
Midterm

The midterm will be held in class on Friday February 27. The duration of the exam is 50 minutes.
 The syllabus includes the material covered in class up until Monday February 23. In particular, this covers all of Chapter 7 and a portion of Chapter 6 of the textbook.
 Here is the list of problems we discussed in class in Wednesday February 25, with a sketch of the solutions.
 The actual midterm with solutions.
Final exam
 The final exam will be held on Monday April 27, 8:30 am in Buchanan B213. Please bring your student ID.
 Practice problem sheet for the final exam.
 Final exams from previous offerings of the course: